Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(7-14\), use cylindrical coordinates to find the indicated quantity. Volume of the solid bounded above by the plane \(z=y+4\), below by the \(x y\)-plane, and laterally by the right circular cylinder having radius 4 and whose axis is the \(z\)-axis.

Step-by-Step Solution

Answer

The volume is \(64\pi\).

Step 1: Understand the Problem

We need to find the volume of the solid bounded by a plane, the xy-plane, and a cylinder. First, convert all given surfaces into cylindrical coordinates. For cylindrical coordinates, use \(x = r \cos(\theta)\), \(y = r \sin(\theta)\), and \(z = z\).

Step 2: Transform Surface Equations to Cylindrical Coordinates

The plane \(z = y + 4\) transforms to \(z = r \sin(\theta) + 4\). The cylinder's equation, given by \(x^2 + y^2 = 16\), transforms to \(r = 4\).

Step 3: Identify Bounds for Integration

Identify the limits for each variable: \(r = [0, 4]\) (from the cylinder), \(\theta = [0, 2\pi]\) (full rotation around the z-axis), and \(z = [0, r \sin(\theta) + 4]\) (bounded below by the xy-plane and above by the plane).

Step 4: Set Up the Integral

The volume \(V\) is calculated by integrating \(1\) over the specified bounds. Use the triple integral:\[V = \int_0^{2\pi} \int_0^4 \int_0^{r \sin(\theta) + 4} 1 \; dz \; r \; dr \; d\theta\]

Step 5: Perform the Integration with Respect to z

Integrate with respect to \(z\):\[\int_0^{r \sin(\theta) + 4} 1 \; dz = \left. z \right|_0^{r \sin(\theta) + 4} = r \sin(\theta) + 4\]

Step 6: Integrate with Respect to r

Now, integrate with respect to \(r\):\[\int_0^4 (r \sin(\theta) + 4) r \; dr = \int_0^4 (r^2 \sin(\theta) + 4r) \; dr\]Solve:\[= \left. \left( \frac{r^3}{3} \sin(\theta) + 2r^2 \right) \right|_0^4 = \left( \frac{64}{3} \sin(\theta) + 32 \right)\]

Step 7: Integrate with Respect to θ

Finally, integrate with respect to \(\theta\):\[\int_0^{2\pi} \left( \frac{64}{3} \sin(\theta) + 32 \right) \; d\theta = \left. \left( -\frac{64}{3} \cos(\theta) + 32 \theta \right) \right|_0^{2\pi}\]Resulting in:\[= \left( -\frac{64}{3} \cos(2\pi) + 32(2\pi) \right) - \left( -\frac{64}{3} \cos(0) + 32(0) \right) = 64\pi\]

Step 8: Conclusion: Calculate the Volume

After evaluating the integral, we find that the volume of the solid is \(64\pi\).

Key Concepts

Volume of Solids

When dealing with volume in mathematics, especially in the context of three-dimensional spaces, it’s important to understand how to calculate it for solids with complex boundaries.

• To determine the volume of such a solid bounded by various surfaces, we need a systematic method, which often involves the use of integration.
• In this exercise, the solid is bounded by a plane, the xy-plane, and a cylinder. Knowing these boundaries will guide us in setting up the necessary integral to find the volume.

Describing these boundaries entails converting them into appropriate coordinate systems, in this case, cylindrical coordinates. This system is especially useful when dealing with objects that have symmetry around an axis, like in this problem. By setting relevant bounds and integrating over the region they enclose, we can derive the volume of even the most irregular solids.

Triple Integrals

Triple integrals are a powerful tool for finding volumes. They extend the concept of integration from one dimension (length) and two dimensions (area) to three dimensions (volume).

• A triple integral allows us to sum up infinitely small volumes (or other quantities) over a three-dimensional region.
• In our problem, we set up a triple integral over a cylindrical boundary to calculate the volume of the solid.

The general form of a triple integral is \[ V = \int \int \int \text{{volume element}} \], where the limits of integration are determined by the boundaries of the solid. In this exercise, the bounds were defined by the cylinder’s radius, the angle of rotation, and the plane equation in cylindrical coordinates. By integrating with respect to \(z\), \(r\), and \(\theta\) in succession, we summed up the contributions to volume from across the entire solid. This step-by-step process ensures no part of the volume is overlooked.

Cylindrical Integration

Cylindrical integration is an integration technique specifically suited for solving problems where cylindrical symmetry is present.

• This system is especially useful in problems dealing with circles, rotations around an axis, or anything resembling a cylinder or cone.
• It involves the transformation of regular Cartesian coordinates into cylindrical coordinates \((r, \theta, z)\).

In cylindrical coordinates:- \(r\) represents the radius from the z-axis.- \(\theta\) is the angle of rotation around the z-axis.- \(z\) stays the same as in Cartesian coordinates.For our problem, the transformation allowed us to describe the cylinder and plane easily. The cylindrical form of integration partitioned the solid into manageable pieces, making it straightforward to integrate over the specified bounds of \(r\), \(\theta\), and \(z\). This approach simplifies working with solids that spread radially and enhances computational efficiency.

Coordinate Transformation

Coordinate transformation is an essential mathematical tool for simplifying problems by shifting from one coordinate system to another.

• This technique allows us to exploit the symmetry and characteristics of the object we are examining.
• For the given problem, transforming from Cartesian to cylindrical coordinates made integration more feasible due to the cylindrical nature of the solid.

In cylindrical coordinates:- Cartesian coordinates \((x, y, z)\) are transformed using \(x = r \cos(\theta)\), \(y = r \sin(\theta)\), and \(z = z\).Changing the coordinates helps to express the problem’s boundaries in a form that suits the objects’ natural geometric properties. It also often reduces the complexity of integration by aligning the coordinate axes with the symmetry of the solid. This transformation was crucial in describing the limits of the triple integral used to find the volume of the solid in our problem efficiently.